A cube with 2.0-cm sides is made of material with a bulk modulus of 4.7 x 10⁵ N/m².When it is subjected to a pressure of 2.0 x 10⁵ Pa the length in cm of its any of its any of its sides is: A)0.85 cm B)1.15 cm C)1.66 cm D)2.0 cm E)none of these

Question

Quizplus · Accepted Answer

We can use the equation for bulk modulus to solve this problem:

Bulk modulus = (pressure change)/(fractional volume change)

Since the cube is subjected to a pressure of 2.0x10^10 Pa, the pressure change is also 2.0x10^10 Pa.

The fractional volume change can be determined from the fact that the cube is isotropic, meaning that it has the same properties in all directions. When a cube is squeezed, all of its sides decrease by the same amount. Therefore, the fractional volume change is equal to the fractional length change.

Using the equation for fractional length change due to pressure,

ΔL/L = -V/E

where ΔL is the change in length, L is the original length, V is the change in volume, and E is the Young's modulus (which is equal to three times the bulk modulus for isotropic materials).

Since the cube is isotropic and its sides are all equal, we can choose any side to represent L.

If each side has a length of 2.0 cm, the original volume of the cube is (2.0 cm)^3 = 8.0 cm^3.

If the pressure causes a fractional volume change of ΔV/V, then the new volume is (1 + ΔV/V) times the original volume. The new side length is then the cube root of the new volume divided by the original surface area (which is 6 times the original side length squared).

Thus,

(1 + ΔV/V) * 8.0 cm^3 = (2.0 cm)^3

ΔV/V = -0.375

Using the equation for fractional length change, we get

ΔL/L = -V/E = -(-0.375)/(3*4.7x10^10 N/m^2) = 2.128x10^-12

The change in length is therefore

ΔL = (2.0 cm)(2.128x10^-12) = 4.256x10^-12 cm

The final length of any side is the original length plus the change in length, so

Final length = 2.0 cm + 4.256x10^-12 cm = 2.00000000213 cm

This is closest to choice (C), 1.66 cm.

A Cube with 2